 |
Business | Industries | Finance | Tax |
| Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |
|
|||||
| First Prev [ 1 2 ] Next Last |
This means that for every complex number z, we have
Since the function f is not injective in (−∞, 0), the function W is multivalued in [-1/e, 0). If we restrict to real arguments x ≥ -1/e and demand w ≥ -1, then a single valued function W0(x) is defined, whose graph is shown. We have W0(0) = 0 and W0(−1/e) = −1.
The Lambert W function cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials and also occurs in the solution of time-delayed differential equations, such as y'(t) = a y(t − 1).
By implicit differentiation, one can show that W satisfies the differential equation
The Taylor series of W0 around 0 can be found using the Lagrange inversion theorem and is given by
The radius of convergence is 1/e, as may be seen by the ratio test. The function defined by this series can be extended to a holomorphic functionHolomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. This is a much stronger condition than r defined on all complex numbers except the real intervalTopology In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. Interval notation is where the permitted values for a variable are expressed as ranging over an in (-∞, -1/e]; this holomorphic function is also called the principal branch of the Lambert W function.
Many equationAlgebra This article is about equations in mathematics. For equations in chemistry, see chemical equation. In mathematics, one often (not quite always) distinguishes between an identity which is an assertion that two expressions are equal regardless of ths involving exponentials can be solved using the W function. The general strategy is to move all instances of the unknown to one side of the equation and make it look like x ex, at which point the W function provides the solution. For instance, to solve the equation 2t = 5t, we divide by 2t to get 1 = 5t e-ln(2)t, then divide by 5 and multiply by -ln(2) to get -ln(2)/5 = -ln(2)t e-ln(2)t. Now application of the W function yields −ln(2)t = W(−ln(2)/5), i.e. t = −W(−ln(2)/5) / ln(2).
Similar techniques show that
has solution
or, equivalently,
The function W(x), and many expressions involving W(x), can be integratedIntegration may be any of the following: Usually integration is the construction of an object, a theory, etc. from separate more limited parts. The result is something composite or integral . Racial integration or simply "integration", in United States us using the substitutionIn calculus, the substitution rule is an important tool for finding antiderivatives and integrals. It is the counterpart to the chain rule of differentiation. Suppose f ''x is an integrable function, and φ t is a continuously differentiable function w w = W(x), i.e. x = w ew:
In a pinch (i.e., outside of computer algebra systems), the W function may be evaluated using the recurrence relationIn mathematics, a recurrence relation also known as a difference equation is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. For example (the logistic map): : Some simply defined
given in Corless et al to compute W.
| Politics | Business | Finance |
 |
 |
 |
|